Theor Chem Acc (2001) 105:413±421
DOI 10.1007/s002140000204
Regular article
Harmonic frequency scaling factors for Hartree-Fock, S-VWN,
B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ
electric property basis set
Mathew D. Halls, Julia Velkovski, H. Bernhard Schlegel
Department of Chemistry, Wayne State University, Detroit, MI 48202, USA
Received: 22 May 2000 / Accepted: 30 August 2000 / Published online: 28 February 2001
Ó Springer-Verlag 2001
Abstract. Scaling factors for obtaining fundamental
vibrational frequencies from harmonic frequencies
calculated at six of the most commonly used levels of
theory have been determined from regression analysis
for the polarized-valence triple-zeta (pVTZ) Sadlej
electric property basis set. The Sadlej harmonic frequency scaling factors for ®rst- and second-row molecules
were derived from a comparison of a total of 900
individual vibrations for 111 molecules with available
experimental frequencies. Overall, the best performers
were the hybrid density functional theory (DFT) methods, Becke's three-parameter exchange functional with
the Lee±Yang±Parr ®t for the correlation functional (B3LYP) and Becke's three-parameter exchange functional
with Perdew and Wang's gradient-corrected correlation
functional (B3-PW91). The uniform scaling factors for
use with the Sadlej pVTZ basis set are 0.9066, 0.9946,
1.0047, 0.9726, 0.9674 and 0.9649 for Hartree±Fock, the
Slater±Dirac exchange functional with the Vosko±Wilk±
Nusair ®t for the correlation functional (S-VWN),
Becke's gradient-corrected exchange functional with
the Lee±Yang±Parr ®t for the correlation functional
(B-LYP), B3-LYP, B3-PW91 and second-order Mùller±
Plesset theory with frozen core (MP2(fc)), respectively.
In addition to uniform frequency scaling factors, dual
scaling factors were determined to improve the agreement between computed and observed frequencies.
The scaling factors for the wavenumber regions below
1800 cm)1 and above 1800 cm)1 are 0.8981 and 0.9097,
1.0216 and 0.9857, 1.0352 and 0.9948, 0.9927 and
0.9659, 0.9873 and 0.9607, 0.9844 and 0.9584
for Hartree±Fock, S-VWN, B-LYP, B3-LYP, B3-PW91
and MP2(fc), respectively. Hybrid DFT methods along
with the Sadlej pVTZ basis set provides reliable theoretical vibrational spectra in a cost-e€ective manner.
This article is supplemented by an internet archive which can be
obtained electronically from the Springer-Verlag server located at
http://link.springer.de/journals/tca
Correspondence to: H. B. Schlegel
e-mail: [email protected]
Key words: Vibrational frequencies ± Scaling factors ±
Density functional theory ± Sadlej basis set
1 Introduction
Theoretical calculations of the vibrational frequencies
and spectral intensities for polyatomic species have
become an invaluable chemical tool over the past
decade. This can be largely attributed to the increased
availability and eciency of robust energy derivative
programs, in which the ®rst and second derivatives are
computed analytically [1], and the increased power of
a€ordable computers. These advances have pushed the
applicability of ®rst-principle investigation of molecular
vibrational properties from the small-molecule domain
to the treatment of reasonably large systems [2±6].
Compared to conventional ab initio methods that
include electron correlation, the superior performance
and cost-e€ectiveness of density functional theory
(DFT) is particularly desirable for larger molecules
[7±9] and transition-metal complexes [10, 11]. The
interpretation of experimental vibrational spectra for
large molecules is greatly assisted by quantum chemical
predictions, particularly in the region below 1800 cm)1,
where the high density of states results in spectral
congestion.
Observed bands can be assigned on the basis of
agreement between computed harmonic frequencies and
experimental frequencies; however, more reliable interpretations can be made by making use of theoretical
intensities as well, which can resolve the assignment of
closely spaced bands. Therefore, it is desirable for a
theoretical method to provide both reliable intensities
and vibrational frequencies.
Previous work comparing theoretical harmonic
frequencies with observed fundamentals have shown that
the calculated frequencies generally overestimate fundamental frequencies, because of the incomplete treatment
of electron correlation, neglect of mechanical anharmonicity and basis set truncation e€ects. To improve the
agreement between the predicted and observed frequencies, the computed harmonic frequencies are usually
414
scaled for comparison. In the ®rst use of scaling, separate
factors were applied to stretching and bending force
constants [12]. A systematic procedure with multiple
scale factors was employed by Blom and Altona [13].
Various scaling strategies have been devised since these
initial e€orts [14±19]. The most sophisticated schemes
developed by Rauhut and Pulay [18] and Baker et al. [19]
use about a dozen parameters to scale force constants
in internal coordinates. Simple uniform scaling achieves
comparable accuracy (about 50% larger root-meansquare error), avoids any ambiguities related to the
choice of internal coordinate system and can be applied
to systems with partial bonding, such as transition states
and clusters. This strategy is adopted in the present work.
A uniform scaling factor of 0.89 was found by
Pople et al. [20] in a comparison of harmonic frequencies computed at the Hartree±Fock/3-21G level of
theory to observed fundamentals. Subsequent studies
comparing Hartree±Fock frequencies computed using
the larger 6-31G(d) basis set and experimental fundamentals suggest very similar uniform scaling factors of
0.8929 and 0.8953 [21, 22]. DFT consistently predicts
harmonic vibrational frequencies in better agreement
with observed fundamentals than conventional ab
initio methods. In comparisons of observed fundamentals with computed harmonics using the 6-31G(d)
basis set, Scott and Radom [23] and Wong [24]
obtained scaling factors of 0.9833 for the Slater±Dirac
exchange functional with the Vosko±Wilk±Nusair ®t
for the correlation functional (S-VWN), 0.9945 for
Becke's gradient-corrected exchange functional with
the Lee±Yang±Parr ®t for the correlation functional
(B-LYP) and 0.9614 for Becke's three-parameter exchange functional with the Lee±Yang±Parr ®t for the
correlation functional (B3-LYP), and for conventional
ab initio methods they found scale factors of 0.9427
and 0.9537 for second-order Mùller±Plesset (MP2) and
quadratic con®guration interaction with single and
doubles excitations (QCISD), respectively.
The theoretical prediction of vibrational intensities is
a computationally demanding task, because of the need
to represent the tail region of the molecular wavefunction [25, 26], which requires medium- to large-sized basis
sets with an adequate complement of polarization and
di€use functions, and because of the need to include
electron correlation. Inclusion of electron correlation
was found to be essential for obtaining quantitative
IR intensities [27±29]. Work in our laboratory has
shown that IR intensities furnished by the hybrid DFT
methods (B3-LYP and Becke's three-parameter exchange
functional with Perdew and Wang's gradient-corrected
correlation functional, B3-PW91) were in closest
agreement with intensities obtained using a highly
correlated ab initio method, QCISD [30]. Good IR
intensities were achieved using medium-sized polarized
basis sets augmented with a set of di€use functions.
Overall the quality of IR intensities obtained by various methods were resolved into the following order:
hybrid DFT > MP2 > local DFT » gradient-corrected
DFT > Hartree±Fock.
The prediction of quantitative Raman intensities is
more costly than the prediction of IR intensities
(which require second derivatives) because they depend
on the square of the polarizability derivative, which
requires calculation of the third derivative of the system energy with respect to geometric coordinates and
the external electric ®eld (¶a/¶Qk ˆ ¶3E/¶Qk¶Fi¶Fk).
Other work in our laboratory has found that the basis
set dependence of predicted Raman intensities was
quite large, requiring signi®cantly larger basis sets to
obtain quantitative results than needed for IR intensities [31]. Raman vibrational intensities were found to
be relatively insensitive to correlation e€ects at levels
of theory applicable to large molecules, in marked
contrast to the observed behavior of theoretical IR
intensities. The quality of the basis set employed was
found to be the dominant consideration in obtaining
quantitative Raman intensities. We found, however,
that the medium-sized Sadlej polarized-valence triplezeta (pVTZ) electric property basis set [32±34],1 which
was ®t to reproduce polarizabilities, provided excellent
quantitative Raman intensities comparable to those
obtained using the much larger Dunning correlation
consistent aug-cc-pVTZ basis set [35±37].
Harmonic vibrational scaling factors for the most
widely used theoretical methods using the Sadlej basis
would be highly bene®cial because of the excellent
quantitative vibrational intensities provided by the Sadlej pVTZ basis set and the substantial savings it a€ords
in computational cost. In the present work, harmonic
vibrational frequencies were computed for a set of 900
individual vibrations for a diverse set of 111 ®rst- and
second-row molecules and were compared to experimental fundamental frequencies [38].2 In this work
we report harmonic frequency scaling factors computed
by regression analysis, for the conventional ab initio
methods, Hartree±Fock and MP2 and the local, gradient-corrected and hybrid density functional methods,
S-VWN, B-LYP, B3-LYP and B3-PW91 using the Sadlej
polarized basis set.
2 Methods
The calculations in this study were performed using the Gaussian
98 suite of programs [39]. Following full geometry optimizations at
each level of theory, harmonic frequencies were computed analytically for Hartree±Fock, MP2 (fc) [40, 41] and the various DFT
methods employed in this study. The vibrational properties were
computed for the test molecules in their ground state, enforcing
their respective point group symmetry. The DFT methods
employed here consisted of the S-VWN local functional [42, 43],
the B-LYP gradient-corrected functional [44, 45] and two hybrid
functionals: B3-LYP [46] and B3-PW91 [47]. The calculations were
carried out with Sadlej's pVTZ basis set, optimized for electric
properties [5s,3p,2d/7s,5p,2d/3s,2p] [32±34]. Complete tables of
numerical data are available on the World Wide Web [48].
1
Sadlej's polarized electric property basis set was obtained from the
Extensible Computational Chemistry Environment Basis Set Database, version 1.0. Paci®c Northwest Laboratory, P.O. Box 999,
Richland, WA 99352, USA.
2
F1 set of molecules used previously by Pople et al. [22], Scott and
Radom [23] and Wong [24] minus AlCl3, BF3, B2H6, BH3CO, CH,
NH, OH, SH, SiH, PH and LiF.
415
3 Discussion and results
When predicting molecular properties using quantum
chemical methods, researchers are often faced with a
trade-o€ between the computational cost and the quality
of results furnished by the method employed. The
development and the use of scaling factors for theoretical harmonic frequencies have greatly extended the
practical utility of results obtained from computationally inexpensive methods. The majority of previous
studies related to the present work have focused on
scaling factors for a number of theoretical methods
using the reasonably small Pople split-valence doublezeta basis set with one set of polarization functions on
heavy atoms, 6-31G(d) [49±51]. Experience has shown
that this basis set is an economical choice yielding
reliable molecular geometries and harmonic frequencies
that, after scaling, provide good agreement with observed fundamentals. In addition to vibrational frequencies, the theoretical prediction of complete vibrational
spectra requires reliable intensities as well. Studies
investigating the basis set dependence of theoretical
vibrational intensities, both IR and Raman, have shown
that intensities computed using the 6-31G(d) and
comparably sized basis sets provide only qualitative
agreement with experimental observations [26]. Recently
we have shown the moderately sized Sadlej polarized
electric property basis set to be an outstanding coste€ective performer in the prediction of quantitative
vibrational intensities [31]. In order to increase the utility
of the Sadlej basis set we determined uniform scaling
factors and low- and high-frequency scaling factors for
the most often used theoretical methods. The molecules
comprising the test set for the work described here are
shown, along with their respective point-group symmetries in Table 1. Together, they provide a set of 900
experimentally well established fundamental frequencies
against which the theoretical harmonic frequencies can
be tested.
3.1 Hartree±Fock
Harmonic frequencies computed at the Hartree-Fock
level are known to systematically overestimate fundamental frequencies by 10±15%. There is good agreement
in the literature de®ning a uniform harmonic frequency
scaling factor for the Hartree±Fock/6-31G(d) level
of theory of 0.8929±0.8953. The Sadlej pVTZ electric
property basis set is roughly twice the size of the Pople 631G(d) basis set. The basis set dependence of theoretical
harmonic frequencies is known to be modest in comparison to that of IR and Raman intensities [52]. In
previous work, the Sadlej basis was shown to furnish
vibrational intensities comparable to those obtained at
around 10 times the computational cost, making the
Sadlej basis an excellent compromise between cost and
performance [31].
Table 1. Set of molecules and respective symmetry used in this study
Molecule
Symmetry
Molecule
Symmetry
NClF2
ClF3
HOCl
SiH3Cl
ClNO
ClNO2
ClNS
NCl2F
SiH2Cl2
Cl2O
SOCl2
Cs
C2v
Cs
C3v
Cs
C2v
Cs
Cs
C2v
C2v
Cs
SCl2
SiHCl3
PCl3
HOF
SiH3F
ONF
NSF
F2NH
N2F2
F2O
C2v
C3v
C3v
Cs
C3v
Cs
Cs
Cs
C2h
C2v
F2SO
Cs
S2F2
SiHF3
NF3
PF3
HNO3
HN3
C2
C3v
C3v
C3v
Cs
Cs
CH3F
C3v
CH4
Td
CH3OH
Cs
CH3NH2
Cs
CH3SiH3
C3v
C2Cl2
D*h
C2N2
D*h
HCCCl
C*v
HCCF
C*v
HCCH
D*h
Trans
C2h
CHClCHCl
Cis CHClCHCl C2v
CH2CCl2
C2v
Cis CHFCHF C2v
Trans CHFCHFC2h
OCHCHO
C2h
CH3CN
C3v
CH3NC
C3v
CH3COF
Cs
CH2CH2
D2h
Trans
C2h
CH2ClCH2Cl
Gauche
C2
CH2ClCH2Cl
c-C2H4O
C2v
CH3CHO
Cs
HCOOCH3
Cs
CH3COOH
Cs
SiH3CCH
C3v
CH3CH2Cl
Cs
Molecule
Symmetry
Molecule
Symmetry
H2O
H2O2
H2S
H2S2
NH3
PH3
SiH4
NO2
N2O
SO2
O3
C2v
C2
C2v
C2
C3v
C3v
Td
C2v
C*v
C2v
C2v
CH3CH2F
c-C2H4NH
CH3CH3
CH3NNCH3
CH3OCH3
CH2CCHCl
HCCCH2Cl
HCCCH2F
CH2CCH2
CH3CCH
CH2CHCHO
Cs
Cs
D3d
C2h
C2v
C1
Cs
Cs
D2d
C3v
Cs
SO3
COClF
ClCN
COCl2
CSCl2
FCN
COF2
CSF2
COS
CO2
D3h
Cs
C*v
C2v
C3v
C*v
C2v
C2v
C*v
D*h
CH3CH2CN
c-C3H6
CH3COCH3
HCCCCH
CH2CHCHCH2
CH3CCH3
H2
HF
N2
O2
Cs
D3h
C2v
D*h
C2h
D3h
D*h
C*v
D*h
D*h
CS2
D*h
F2
D*h
CHCl3
CHF3
HCN
HNCO
CH2Cl2
H2CO
C3v
C3v
C*v
Cs
C2v
C2v
CH2(1A1)
CH2(3B1)
HCO
HCOOH
CH3Cl
C2v
C2v
Cs
Cs
C3v
416
The average di€erence, the average absolute di€erence and the standard deviation from experimental
data are presented in Table 2. Not surprisingly, the
Hartree±Fock frequencies are in worse agreement with
experimental fundamentals than the other theoretical
methods considered here. This is expected, because of
the neglect of electron correlation in the Hartree±Fock
calculations. An important observation for the results
presented in Table 2 is that the average di€erence and
the average absolute di€erence are almost identical
(within 3 cm)1) giving excellent motivation to develop a
uniform scaling factor for this level of theory. Leastsquares regression analysis gives a suggested Hartree±
Fock frequency scaling factor of 0.9066. This scaling
factor is a little larger than the scaling factors determined for the 6-31G(d) basis set, but is comparable
to scaling factors determined for larger basis sets such
as 6-311G(d,p) and 6-311G(2d,p) of 0.9051 and 0.9054,
respectively [23]. After scaling, the Hartree±Fock/Sadlej
frequencies are in much better agreement, showing a
marked decrease in the average di€erence, the average
absolute di€erence and the standard deviation of
around 145, 112 and 54 cm)1, respectively. The e€ect
of uniform scaling on the error in the Hartree±Fock
frequencies is shown in Fig. 1. The agreement for unscaled frequencies is severely biased toward overestimation. Upon uniform scaling the distribution of errors
becomes much more satisfactory, peaking around zero
di€erence. Some of the problematic molecules for
which the scaled Hartree±Fock/Sadlej frequencies
were in worst agreement are NSF, F2O, F2 and O3.
Scott and Radom [23] have previously noted severe
disagreement between Hartree±Fock frequencies and
experimental fundamentals for these systems.
Given the overestimation of vibrational frequencies
at the Hartree±Fock level, the inclusion of electron
correlation, even at minimal levels, should improve the
agreement between computed harmonic frequencies and
experimental fundamentals. It has been noted that the
e€ect of electron correlation on molecular geometries
is to slightly elongate bonds, leading to a systematic
decrease in force constants and, therefore, lower
vibrational frequencies [53±55].
Table 2. Scaling factors and
average di€erence, average absolute di€erence and standard
deviation from experimental
fundamentals with the Sadlej
polarized-valence triple-zeta
basis set (cm)1)
Raw frequencies
Average di€erence
Average absolute di€erence
Standard deviation
Uniform scaling
Uniform scaling factor
Average di€erence
Average absolute di€erence
Standard deviation
Dual scaling
<1800 cm)1 scaling factor
>1800 cm)1 scaling factor
Average di€erence
Average absolute di€erence
Standard deviation
3.2 Local and gradient-corrected DFT
Over the past decade, DFT has become an extremely
popular alternative to conventional correlated ab initio
methods. DFT methods provide molecular properties
and energetics in similar or better agreement with
experimental data than methods such as MP2. Previous
studies examining the performance of local and gradientcorrected DFT in computing vibrational frequencies
have shown that they provide good agreement with
observed fundamental frequencies. The uniform scaling
factors for the S-VWN and B-LYP levels of theory with
the 6-31G(d) basis set have been determined to be 0.9833
and 0.9945±0.995, respectively [23, 24, 56]. Both scaling
factors are signi®cantly closer to unity than that of
Hartree±Fock, suggesting that the theoretical harmonic
frequencies obtained by these methods may be used
without scaling. The di€erences and the standard deviation for the S-VWN/Sadlej and the B-LYP/Sadlej levels
of theory are shown in Table 2. In contrast to Hartree±
Fock, comparisons of the average di€erence and the
average absolute di€erence for these methods show that
the predicted harmonic frequencies are distributed
around the experimental fundamental frequencies, with
S-VWN overestimating the frequencies more than
B-LYP. Also both S-VWN and B-LYP have standard
deviations roughly half that of Hartree±Fock. Regression analysis gives uniform scaling factors of 0.9946 and
1.0047 for S-VWN/Sadlej and B-LYP/Sadlej, respectively. As with Hartree±Fock, the scaling factors determined
for the Sadlej basis set for S-VWN and B-LYP are
slightly larger than those for the 6-31G(d) basis set.
Scaling factors for S-VWN frequencies obtained using a
larger basis set than 6-31G(d) have not been reported;
however, for the B-LYP/cc-pVDZ level of theory, a
uniform scaling factor of 1.00 has been suggested [57], in
agreement with work presented here. After uniform
scaling the predicted frequencies are better distributed
around the experimental values; however, the e€ect is
small, as expected with scaling factors so close to unity.
Histograms of the errors in the S-VWN/Sadlej and the
B-LYP/Sadlej frequencies are shown in Figs. 2 and 3,
respectively. An immediate observation is that compared
Hartree±Fock S-VWN
B-LYP
B3-LYP
B3-PW91
MP2(fc)
146.66
149.39
109.27
)26.61
42.37
47.85
25.87
44.69
59.53
34.64
49.12
63.68
33.09
55.77
88.79
)6.45
44.17
54.66
0.9066
1.99
36.96
55.71
0.9946
)14.05
43.10
52.40
1.0047
)20.08
41.49
50.31
0.8981
0.9097
)2.01
37.45
54.88
1.0216
0.9857
)2.61
35.21
47.32
1.0352
0.9948
)7.47
30.97
44.95
0.9726
0.9674
)13.27
)12.30
33.04
33.67
42.01
42.91
0.9927
0.9659
)4.60
25.23
38.84
0.9873
0.9607
)3.63
26.29
39.64
0.9649
)17.25
41.73
70.03
0.9844
0.9584
)8.84
31.84
68.82
417
with Hartree±Fock/Sadlej the distribution of errors is
much broader, but with fewer large errors, limiting the
improvement that can be obtained by uniform scaling. SVWN seems to underestimate vibrational frequencies as
often and in similar magnitude as overestimate them and
the e€ect of uniform scaling is very small. B-LYP tends to
underestimate vibrational frequencies and uniform scaling has some e€ect in shifting the error distribution closer
to zero. Some of the problematic molecules for which the
scaled S-VWN/Sadlej and B-LYP/Sadlej frequencies
were in poor agreement are NSF and NSCl.
3.3 Hybrid DFT
Previous results obtained using hybrid DFT methods
and the 6-31G(d) basis set have suggested that the
hybrid DFT methods are the most successful procedures for predicting vibrational frequencies in agreement with experimental fundamentals. In addition to
providing harmonic frequencies that can be easily
scaled for reliable prediction of fundamentals, hybrid
DFT has been shown to outperform Hartree±Fock,
local and gradient-corrected DFT and MP2 in predict-
Fig. 1. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
Hartree±Fock/Sadlej pVTZ level of theory
Fig. 2. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
S-VWN/Sadlej pVTZ level of theory
418
Fig. 3. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
B-LYP/Sadlej pVTZ level of theory
ing IR and Raman intensities [30, 31]. The uniform
scaling factors for the B3-LYP/6-31G(d) and B3-PW91/
6-31G(d) levels of theory have been determined to be
0.9614 and 0.9573, respectively [23]. The average
di€erence, the average absolute di€erence and the
standard deviation for the B3-LYP and B3-PW91
methods are shown in Table 2. Comparing the average
di€erence and the average absolute di€erence for these
methods suggests that the predicted harmonic frequencies are distributed around the experimental numbers,
with some tendency for overestimation, behavior intermediate between that observed for Hartree±Fock and
the pure DFT methods. The uniform scaling factors for
B3-LYP/Sadlej and B3-PW91/Sadlej have been determined to be 0.9726 and 0.9674, respectively, similar to,
but a little larger than, those reported with the 6-31G(d)
basis set. This new scaling factor for B3-LYP/Sadlej
replaces our earlier estimate of 0.9663, computed for a
much smaller set of test molecules [31]. Uniform scaling
signi®cantly decreases the average di€erences, the
average absolute di€erences and the standard deviation
for both methods. The e€ect of uniform scaling on the
error distribution for the B3-LYP/Sadlej and B3-PW91/
Sadlej frequencies can be seen in Figs. 4 and 5,
respectively. The raw hybrid DFT frequencies have
many large overestimation errors, similar to that
observed for Hartree±Fock. Uniform scaling does an
excellent job in diminishing the number of extreme
outliers; however, the scaled distribution peaks just
under zero, showing a tendency of the uniformly scaled
frequencies to slightly underestimate fundamentals. As
with the Hartree±Fock and the pure DFT methods,
hybrid DFT calculations on the molecules NSF and
NSCl showed some of the largest di€erences between
theory and experiment.
3.4 Second-order Mùller±Plesset theory
MP2 is the only conventional correlated ab initio
method for which calculations on larger-sized chemical
systems are feasible. Previous studies assessing the
agreement between experiment and MP2 frequencies
using the 6-31G(d), 6-31G(d,p) and 6-311G(d,p) basis
sets have suggested uniform scaling factors of 0.9427±
0.9434, 0.9370 and 0.9496, respectively [21±23]. The
average di€erence, the average absolute di€erence and
the standard deviation from experimental fundamentals
are listed for MP2 in Table 2. The MP2 di€erences are
very similar to those obtained using the hybrid DFT
methods; however, the standard deviation is larger than
for those methods. Regression analysis yields a uniform
scaling factor of 0.9649 for the MP2/Sadlej level of
theory, also comparable to the scaling factors determined for the hybrid DFT methods and a little closer to
unity than scaling factors determined for MP2 using the
smaller Pople basis sets. In previous studies using the 631G(d) basis set [21±23], similar behavior was observed
for MP2 frequencies. The e€ect of scaling on the error
distribution for MP2/Sadlej frequencies is shown in
Fig. 6. The distribution of errors shown there resembles
those for the hybrid methods described previously, with
uniform scaling diminishing the number of extreme
outliers and the scaled distribution peaks just under
zero, showing a tendency of a uniformly scaled frequencies to slightly underestimate fundamentals. Even after
uniform scaling the MP2 standard deviation seems
suspiciously high in comparison to the DFT methods
considered here. The apparent discrepancy in the
in¯ated standard deviation is resolved by noting the
e€ect of a few severe outliers. Pople and coworkers [22,
58] have pointed out that MP2 theory fails to adequately
419
Fig. 4. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
B3-LYP/Sadlej pVTZ level of theory
Fig. 5. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
B3-PW91/Sadlej pVTZ level of theory
describe O3 and NO2 asymmetric stretching frequencies.
If the most severe outliers (O3, NO2, NSF and NSCl) for
the MP2/Sadlej frequencies are neglected, the uniform
scaling factors are practically unchanged, whereas the
standard deviation decreases from around 70 to 49 cm)1.3
3
MP2(fc) calculations on C2Cl2 with the Sadlej basis set resulted in
one imaginary frequency. Subsequent calculations using the
6-31G(d) basis set also gave one imaginary frequency, suggesting
an inherent weakness in MP2 theory to adequately describe
this molecule.
3.5 Dual scaling
Experimental vibrational spectra are usually discussed
in terms of di€erent wavenumber regions known to
generally correspond to di€erent types of vibrations.
The upper wavenumber region, above 1800 cm)1,
contains vibrations composed largely of localized hydrogen stretches, whereas the mid wavenumber region
contains heavy atom in-plane stretches and bends,
and the low wavenumber region, the out-of-plane and
torsional modes. It is in the latter two regions, below
420
Fig. 6. Histogram of frequency di€erences
between computed harmonics and observed
fundamentals for the unscaled and scaled
MP2/Sadlej pVTZ level of theory
1800 cm)1 (the ®ngerprint region), where quantum
chemical prediction can be the most useful in making
vibrational band assignments that may not be otherwise
interpretable. Also, high-energy modes can be expected
to be more anharmonic, leading to greater errors
because of the harmonic approximation. To investigate
the utility of separate scaling factors (dual scaling)
for the two ranges <1800 cm)1 and >1800 cm)1, we
reanalyzed the agreement between the theoretical harmonic frequencies and the experimental fundamentals
for these two ranges. The dual scaling factors for the
levels of theory considered here using the Sadlej pVTZ
basis set are listed in Table 2. An interesting ®rst
observation is in comparing dual scaling factors: the
two pure DFT methods, S-VWN and B-LYP, tend
to underestimate the low-frequency vibrations and
tend to overestimate the high-frequency vibrations.
The di€erence between the <1800 cm)1 and the
>1800 cm)1 scale factors are approx. +3% for correlated methods and approx )1% for Hartree±Fock. The
dual scaling procedure does an excellent job in improving the error distribution between the scaled theoretical
frequencies and experimental fundamental frequencies.
The dual scaled error distributions are peaked around
zero and the distributions are more symmetric, compared with the raw and uniform scaled frequency errors.
The Hartree±Fock agreement is the least a€ected by
dual scaling, whereas the DFT methods and MP2 share
increased agreement with experiment. The e€ect of dual
scaling on the error distributions is evident from
Figs. 1±6. All methods show an improved error distribution from the raw and uniform scaled data. In
particular, the histograms for the hybrid DFT methods,
B3-LYP and B3-PW91, and MP2 show a marked
improvement using dual scaling.
4 Conclusions
The purpose of this work was to determine harmonic
frequency scaling factors for use with the Sadlej electric
property basis set, which is known to provide excellent
vibrational intensities in a cost-e€ective fashion. Uniform scaling factors were determined through leastsquares analysis for the Hartree±Fock, S-VWN, B-LYP,
B3-LYP, B3-PW91 and MP2 levels of theory. In
addition to uniform scaling factors, dual scaling factors
were determined which were shown to favorably increase
the agreement between computed harmonic frequencies
and experimental fundamentals. The hybrid DFT methods, B3-LYP and B3-PW91, were found to be most
reliable for the prediction of vibrational frequencies,
outperforming MP2 at signi®cantly less cost. For the
prediction of vibrational spectra the use of hybrid DFT
methods along with the Sadlej pVTZ basis set is coste€ective and yields excellent results for both frequencies
and intensities.
Acknowledgements. We gratefully acknowledge ®nancial support
from the National Science Foundation (grant no. CHE9874005)
and a grant for computing resources from NCSA (grant no.
CHE980042 N). M.D.H. would also like to thank Wayne State
University for ®nancial support provided by a Thomas C. Rumble
Graduate Fellowship.
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